This is a contributed topic on abstract relational biology and its close connections to the theory of
categories, especially the concrete and abstract categories of sets, Set.
1 Introduction
Abstract relational biology (ARB) is an area of mathematical or theoretical biology in which
networks of linked physiological and biochemical functions of living cells and multi-cellular
organisms are defined over sets Si and their elements ej ∈ Si representing for example
specific metabolic products or other biomolecules; such organismic sets Si are then
assembled in set-related mathematical constructions-such as categories of sets- by making
abstraction of the underlying, physical and (bio) molecular structures that implement the
cellular or physiological functions of a living organism. Thus, the early formulations of
ARB by Nicolas Rashevsky were based on set theory, molecular set theory, and the
classical logic of predicates and logical propositions. Therefore, a natural foundation
for ARB would currently be the modern relation theory. The relational structure of
an organism is strongly emphasized over the anatomical structure and the molecular
structure of cells and other components of the organism such as the chromosomes, genome,
mitochondria, endoplasmic reticulum, membranes, and so on. However, the relational
structure is defined in a general, mathematical sense, such as the relational structure of a
category; the initial choice made by Robert Rosen was that of the category of sets, Set (or
Ens). The relational structure was thus limited, both in terms of the objects’ relational
structure as well as in terms of the relational structures definable by morphisms that were
restricted to set-theoretical mappings, or maps between sets. Subsequent developments
(beginning in 1968) extended ARB to categories with structure, thus not limiting ARB
objects to being sets, or the morphisms to being maps between sets. Furthermore, classes
were introduced instead of sets, thus not limiting the categorical framework to small
categories.
Since 1952 there have been two major, set-based theories in abstract relational biology that are
concisely outlined next.
2 Organismic Set Theory and Abstract–Relational,
Metabolic–Replication, (M,R)–Systems
2.1 Brief history
Two major proponents were Nicolas Rashevsky (up to 1973) who is one of the founders of
mathematical biophysics and mathematical biology, and Robert Rosen, his former PhD student at
the University of Chicago. Nicolas Rashevsky formulated the mathematical theory of organismic
sets (OS) that are organized beginning with the genetic level, continuing to the cellular level, and
then to higher levels of multi-cellular organization, activities and products; his theory was similarly
formulated for societies organized at such different levels. Subsequently, it was shown that
Rashevsky’s organismic sets can be represented in terms of categories of algebraic theories
[3].
Robert Rosen introduced (metabolic–repair) models, or (M,R)-systems in 1957 ([11, 12]);
such systems will be here abbreviated as MR-systems, (or simply MR’s). Rosen, then
represented the MR’s in terms of categories of sets, deliberately selected without any
structure other than the discrete topology of sets. He also considered biocomplexity to
be an emergent, defining feature of organisms which is not reducible in terms of the
molecular structures (or molecular components) of the organism and their physicochemical
interactions.
2.2 Basic ARB concepts
Definition 2.1. The simplest MR-system represents a relational model of the primordial
organism which is defined by the following categorical sequence (or diagram) of sets and
set-theoretical mappings: f : A → B,ϕ : B → HomMR(A,B), where A is the set of inputs
to the MR-system, B is the set of its outputs, and ϕ is defined as the ‘repair map’, or
R-component, of the MR-system which associates to a certain product, or output b, the
metabolic component (such as an enzyme, E, for example) represented by the set-theoretical
mapping f. Then, HomMR(A,B) is defined as the set of all such metabolic components
represented by set-theoretical) mappings f.
(occasionally written incorrectly as 
)
Definition 2.2. A general (M,R)-system was defined by Rosen (1958a,b) as the category
of the metabolic and repair components (that were specified above in Definition 0.1),
which are networked in a complex, abstract ‘organism’ defined by all the abstract relations
and connecting maps between the sets specifying all the metabolic and repair components
of such a general, abstract model of the biological organism. The morphisms of the
(M,R)-system category are the metabolic and repair set-theoretical mappings, such as
f and ϕ, and its objects are the sets Ai,Bi, whereas f ∈ HomMRi(Ai,Bi) and ϕ ∈
HomMRi[B,HomMRi(Ai,Bi)], with i ∈ I, and I being a finite index set, or directed set.
Remarks
With a few, additional notational changes it can be shown that the (M,R)-system category is a
subcategory of the category of automata (or sequential machines; [4, 5]). However, in his last
published book in 1997 on “Essays on Life Itself”, Robert Rosen finally accepted the need for
representing organisms in terms of categories with structure that entail biological functions, both
metabolic and repair ones. Note also that, unlike Rashevsky in his theory of organismic sets, Rosen
did not attempt to extend the MRs to modeling societies, even though with appropriate
modifications, such as the introduction of Rosetta biogroupoid structures ([9, ?]) in generalized
(M,R)-system categories with structure ([3, 8]), this is feasible and yields meaningful
mathematical and sociological results.
References
[1] Rashevsky, N.: 1965, The Representation of Organisms in Terms of Predicates,
Bulletin of Mathematical Biophysics 27: 477-491.
[2] Rashevsky, N.: 1969, Outline of a Unified Approach to Physics, Biology and
Sociology., Bulletin of Mathematical Biophysics 31: 159–198.
[3] Baianu, I.C.: 1970, Organismic Supercategories: II. On Multistable Systems.,Bulletin
of Mathematical Biophysics, 32: 539-561.
[4] Baianu, I.C., Some Algebraic Properties of (M,R)–systems., Bulletin of
Mathematical Biophysics, 35: 318-323.
[5] Baianu, I.C. and M.M. Marinescu. 1974. On a Functorial Construction of
(M,R)–systems. Rev. Roumaine Math., 215-224.
[6] Baianu, I.C.: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras:
The Non-linear Theory., Bulletin of Mathematical Biology, 39: 249-258.
[7] Baianu, I.C.: 1980, Natural Transformations of Organismic Structures., Bulletin of
Mathematical Biology,42: 431-446.
[8] Baianu, I.C.: 1987. Mathematical Models in Medicine, vol. 7., Ch.11 Pergamon Press,
New York, 1513 -1577; URLs: CERN Preprint No. EXT-2004-072:, available here as
PDF, or as as an archived html document.
[9] Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006b,
Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and
Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic
Networks., Axiomathes, 16 Nos. 1–2: 65–122.
[10] Rosen, R.: 1985, Anticipatory Systems, Pergamon Press: New York.
[11] Rosen, R.: 1958a, A Relational Theory of Biological Systems., Bulletin of
Mathematical Biophysics 20: 245-260.
[12] Rosen, R.: 1958b, The Representation of Biological Systems from the Standpoint of
the Theory of Categories., Bulletin of Mathematical Biophysics 20: 317-341.
[13] Rosen, R.: 1987, On Complex Systems, European Journal of Operational Research
30:129–134.
